Lemma 3.2 ([[1]}, Lemma 3.16])
Let \(p: \mathcal {X}\rightarrow S\)  be a family of projective Deligne-Mumford stacks with a moduli scheme \(\pi :\mathcal {X}\rightarrow X\)  and a relative polarization \((\mathcal {E}, \mathcal {O}_{X}(1))\) . Assume that \(S\)  is connected and \(\mathcal {F}\)  is an \(S\) -flat sheaf on \(\mathcal {X}\) . Then there is a polynomial \(P\)  such that for every geometric point \(\mathrm {Spec}\,k\xrightarrow{} S\)  the modified Hilbert polynomial of the fiber \(\chi (\mathcal {X}_{s},\mathcal {F}\otimes \mathcal {E}^\vee \otimes \pi ^*\mathcal {O}_{X}(m)|_{\mathcal {X}_{s}})=P(m)\) .

